Threshold Exceedances - Universität zu Köln

The GPD MethodThe Hill Method

Sources

Threshold Exceedances

Moritz Lücke

27. April 2018

Moritz Lücke Threshold Exceedances


Sources

1 The GPD MethodEstimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk

2 The Hill Method

3 Sources



Sources

Estimating ξ and βEstimating the Thresholdmodelling Tails and Measures of Tail risk



Sources




Sources


Excess Distribution

DefinitionLet X be a rv with df F. The excess distribution over the thresholdu has the df

Fupxq “ PpX ´ u ď x |X ą uq “F px ` uq ´ F puq

1´ F puq

for 0 ď x ă XF ´ u.



Sources


The GPD

DefinitionThe General Pareto Distribution (GPD) is given by:

Gξ,βpxq “

#

1´ p1` xξβ q´ 1ξ ξ ‰ 0

1´ expp´ xβ q ξ “ 0

for β ą 0, x ě 0 if ξ ě 0 and 0 ď x ď ´βξ if ξ ă 0.



Sources


Abbildung: GPD mit ξ “ ´0.5; 0; 0.5 und β “ 1



Sources


TheoremWe can find a positive-measurable function βpuq so that

limuÑXF

sup0ďxăXFú

|Fupxq ´ Gξ,βpuqpxq| “ 0

if and only if F P MDApHξq, ξ P R.



Sources


Use of the Theorem

We assume that for some high threshold u, we haveFupxq “ Gξ,βpxq for 0 ď x ă XF ´ u and some ξ P R and β ą 0.



Sources


Estimating ξ and β

Given the loss data X1, ...,Xn from F , a random number Nu

will exceed our threshold u.

Relabel these X 11, ...,X1Nu .

We write Y 1j “ X 1j ´ u.We can use the log-likelihood method:

Lpξ, β;Y1, ...,YNuq “

Nuÿ

j“1

lnpgξ,βpYjqq

“ Ńu lnpβq ´ p1`1ξq

Nuÿ

j“1

lnp1` ξYj

βq,



Sources




will exceed our threshold u.Relabel these X 11, ...,X

1Nu .



Nuÿ

j“1

lnpgξ,βpYjqq


Nuÿ

j“1

lnp1` ξYj

βq,



Sources





1Nu .

We write Y 1j “ X 1j ´ u.

We can use the log-likelihood method:


Nuÿ

j“1

lnpgξ,βpYjqq


Nuÿ

j“1

lnp1` ξYj

βq,



Sources





1Nu .



Nuÿ

j“1

lnpgξ,βpYjqq


Nuÿ

j“1

lnp1` ξYj

βq,



Sources


TheoremWe can find a positive-measurable function βpuq so that

limuÑXF

sup0ďxăXFú

|Fupxq ´ Gξ,βpuqpxq| “ 0

if and only if F P MDApHξq, ξ P R.



Sources


DefinitionThe mean excess function of an rv X with finite mean is given byepuq “ E pX ´ u|X ą uq.

TheoremUnder the assumption Fupxq “ Gξ,βpxq, it follows that the meanexcess function is linear for all v ą u



Sources


DefinitionThe mean excess function of an rv X with finite mean is given byepuq “ E pX ´ u|X ą uq.

TheoremUnder the assumption Fupxq “ Gξ,βpxq, it follows that the meanexcess function is linear for all v ą u



Sources


For positive-valued loss data X1, ...,Xn we estimate the meanexcess function with the sample mean excess function given by

enpvq “

řni“1pXi ´ vqIpXiąvq

řni“1 IpXiąvq

.



Sources


We construct the mean excess plottpXi ,n, enpXi ,nqq : 2 ď i ď nu where Xi denotes the upper ithorder statistic.

If the data support a GDP model over a high threshold, thenthe plot should become increasingly linear for higher values ofv .By this, we can estimate the needed high of our threshold.



Sources


We construct the mean excess plottpXi ,n, enpXi ,nqq : 2 ď i ď nu where Xi denotes the upper ithorder statistic.If the data support a GDP model over a high threshold, thenthe plot should become increasingly linear for higher values ofv .

By this, we can estimate the needed high of our threshold.



Sources


We construct the mean excess plottpXi ,n, enpXi ,nqq : 2 ď i ď nu where Xi denotes the upper ithorder statistic.If the data support a GDP model over a high threshold, thenthe plot should become increasingly linear for higher values ofv .By this, we can estimate the needed high of our threshold.



Sources




Sources




Sources


modelling Tails and Measures of Tail risk



Sources


Our goal is to estimate the tail of a underlying loss distribution Fand associated risk measures.



Sources


We have for x ě u

F̄ pxq “ PpX ą uqPpX ą x |X ą uq

“ F̄ puqPpX ´ u ą x ´ u|X ą uq

“ F̄ puqF̄upx ´ uq

“ F̄ puqp1` ξx ´ u

βq´ 1ξ

which gives us a formula for tail probabilities, if F puq is known.If not, we can estimate F̄ puq with the estimator Nu

n .

ñ ˆ̄F pxq “ Nun p1` ξ̂

xúβ̂q´ 1ξ̂ .



Sources


We have for x ě u





βq´ 1ξ

which gives us a formula for tail probabilities, if F puq is known.

If not, we can estimate F̄ puq with the estimator Nun .


xúβ̂q´ 1ξ̂ .



Sources


We have for x ě u





βq´ 1ξ


n .


xúβ̂q´ 1ξ̂ .



Sources


We have for x ě u





βq´ 1ξ


n .


xúβ̂q´ 1ξ̂ .



Sources


By inverting the formula, we can obtain a high quantile of theunderlying distribution, which we can interpret as a VaR . Forα ď F puq we have

VaRα “ qαpF q “ u `β

ξpp1´ αF̄ puq

q´ξ ´ 1q.

Assuming ξ ă 1, the associated expected shortfall is given by

ESα “1

1´ α

ż 1

αqxpF qdx “

VaRα1´ ξ

`β ´ ξu

1´ ξ.



Sources




ξpp1´ αF̄ puq

q´ξ ´ 1q.


ESα “1

1´ α

ż 1

αqxpF qdx “

VaRα1´ ξ

`β ´ ξu

1´ ξ.



Sources




ξpp1´ αF̄ puq

q´ξ ´ 1q.


ESα “1

1´ α

ż 1

αqxpF qdx “

VaRα1´ ξ

`β ´ ξu

1´ ξ.



Sources

The Hill Method

Alternative to the GPD Method.

New assumption: F P MDApHξq, ξ ą 0.We can use the Fréchet Theorem(F P MDApHξq ô F̄ “ x´

1ξ Lpxq for ξ ą 0)

ñ F̄ “ x´αLpxq for a function L P R0 and α “ 1ξ ą 0.



Sources

The Hill Method

Alternative to the GPD Method.New assumption: F P MDApHξq, ξ ą 0.

We can use the Fréchet Theorem(F P MDApHξq ô F̄ “ x´





Sources

The Hill Method

Alternative to the GPD Method.New assumption: F P MDApHξq, ξ ą 0.We can use the Fréchet Theorem(F P MDApHξq ô F̄ “ x´





Sources

The Hill Method

Alternative to the GPD Method.New assumption: F P MDApHξq, ξ ą 0.We can use the Fréchet Theorem(F P MDApHξq ô F̄ “ x´





Sources

Estimating α

Given the data X1, ...,Xn, we first build the orderstatisticXn,n ď ... ď X2,n ď X1,n

the hill estimator is then given by

α̂Hk,n “ p

1k

kÿ

j“1

lnpXj ,nq ´ lnpXk,nqq´1

for 2 ď k ď n.



Sources

Estimating α



α̂Hk,n “ p

1k

kÿ

j“1


for 2 ď k ď n.



Sources

Estimating α



α̂Hk,n “ p

1k

kÿ

j“1


for 2 ď k ď n.



Sources

Estimating α



α̂Hk,n “ p

1k

kÿ

j“1


for 2 ď k ď n.

The strategy is to plot Hill estimates for various values of k.This gives the Hill plot (pk, α̂H

k,nq : k “ 2, ..., n). We hope tofind a stable region in the Hill plot.



Sources

Estimating α



α̂Hk,n “ p

1k

kÿ

j“1


for 2 ď k ď n.The strategy is to plot Hill estimates for various values of k.This gives the Hill plot (pk, α̂H

k,nq : k “ 2, ..., n). We hope tofind a stable region in the Hill plot.



Sources

Estimating α



Sources

hill based tail estimates

We assume a tail of the form F̄ pxq “ Cx´α, x ě u ą 0 forsome high threshold u.

We estimate α by α̂pHqk,n and u by Xk,n



Sources

hill based tail estimates

We assume a tail of the form F̄ pxq “ Cx´α, x ě u ą 0 forsome high threshold u.

We estimate α by α̂pHqk,n and u by Xk,n



Sources

estimating C

F̄ puq “ Cu´α̂pHq

k,n

ô C “ uα̂pHq

k,n F̄ puq

The empirical estimator for F̄ puq is kn .

We get the Hill tail estimator

ˆ̄F pxq “k

np

x

Xk,nq´α̂

pHq

k,n, x ď Xk,n.



Sources

estimating C


k,n ô C “ uα̂pHq

k,n F̄ puq



ˆ̄F pxq “k

np

x

Xk,nq´α̂

pHq

k,n, x ď Xk,n.



Sources

estimating C



k,n F̄ puq



ˆ̄F pxq “k

np

x

Xk,nq´α̂

pHq

k,n, x ď Xk,n.



Sources

estimating C



k,n F̄ puq



ˆ̄F pxq “k

np

x

Xk,nq´α̂

pHq

k,n, x ď Xk,n.



Sources

Sources

[1] A.McNeil R.Frey P.Embrechts. Quantitative Risk Management:Concepts, Techniques and Tools. Princeton Series in Finance, 2015.


Threshold Exceedances - Universität zu Köln

Documents